1. (Opposing interets of men and women) Consider an arbitrary marriage problem. Let f be the...
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1. (Opposing interets of men and women) Consider an arbitrary marriage problem. Let f be the men-proposing DA matching of this problem. Let f be an arbitrary stable matching of this problem. We show via the following steps that every woman finds her outcome at f no worse than her outcome at fM. (Similarly, you can show that every man finds his outcome at f no worse than his outcome at women-proposing DA matching.) (a) Suppose to the contrary that woman w₁ finds her outcome at f worse than her outcome at fM. Argue that woman w₁ must not be single at fM. (Hint: f is stable.) (b) Assume that woman w₁ marries man m₁ at fM. Argue that man mi's outcome at f is not to marry woman w₁, i.e., f(m₁) + w₁. (Hint: Woman w₁ is happier at fM.) (c) Use the conclusion in the previous step to argue that man m₁ cannot be indifferent between his outcome at f, f(m₁), and his outcome at fM, w₁. (d) Argue that man m₁ cannot prefer w₁ to f(m₁). (Hint: f is stable). (e) Argue that man m₁ must prefer f(m₁) to w₁. Then find a contradiction to the result we have proved in class, that is, men-proposing DA algorithm is optimal for men. 1. (Opposing interets of men and women) Consider an arbitrary marriage problem. Let f be the men-proposing DA matching of this problem. Let f be an arbitrary stable matching of this problem. We show via the following steps that every woman finds her outcome at f no worse than her outcome at fM. (Similarly, you can show that every man finds his outcome at f no worse than his outcome at women-proposing DA matching.) (a) Suppose to the contrary that woman w₁ finds her outcome at f worse than her outcome at fM. Argue that woman w₁ must not be single at fM. (Hint: f is stable.) (b) Assume that woman w₁ marries man m₁ at fM. Argue that man mi's outcome at f is not to marry woman w₁, i.e., f(m₁) + w₁. (Hint: Woman w₁ is happier at fM.) (c) Use the conclusion in the previous step to argue that man m₁ cannot be indifferent between his outcome at f, f(m₁), and his outcome at fM, w₁. (d) Argue that man m₁ cannot prefer w₁ to f(m₁). (Hint: f is stable). (e) Argue that man m₁ must prefer f(m₁) to w₁. Then find a contradiction to the result we have proved in class, that is, men-proposing DA algorithm is optimal for men.
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Fundamentals of corporate finance
ISBN: 978-0470876442
2nd Edition
Authors: Robert Parrino, David S. Kidwell, Thomas W. Bates
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